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Let K be a subclass of Mod() which is closed under isomorphism. Vaught showed that K is ${\Sigma }_{\alpha }$ (respectively, ${\Pi }_{\alpha }$) in the Borel hierarchy iff K is axiomatized by an infinitary ${\Sigma }_{\alpha }$ (respectively, ${\Pi }_{\alpha }$) sentence. We prove a generalization of Vaught’s theorem for the effective Borel hierarchy, i.e. the Borel sets formed by union and complementation over c.e. sets. This result says that we can axiomatize an effective ${\Sigma }_{\alpha }$ or effective ${\Pi }_{\alpha }$ Borel set with a computable infinitary sentence of the same complexity. This result...